'Any' is different than 'many,' as you originally claimed. I think if the report you seem most focused on was estimating that 2.1% of all rape claims are false accusations, your concern would be more understandable. But this is a paper on rapes reported in a specific geographical area  over a 3-year period and 2.1 is the percentage of reports designated false by the police, making this an odd choice of information to focus in on as telling a "different story." Most groups that compile these data in order to make estimates do place those estimates in ranges, anywhere from 2-3 to 2-10%.

if they are false at a higher rate than cases accuser and police followed through with, the 2.1% is a significant underestimate.

This still seems to suggest that cases resulting in withdrawals or no further action are more likely to be false, and you don't have enough information for that assumption to be well-founded. Further, this assumption feeds into harmful myths about the underlying causes for withdrawal and case attrition in sexual assault reports, many of which are addressed in the sources I've linked above.

Only 2.1% were designated by the cops as false, but that was after the majority of accusations had been withdrawn or determined 'no further action'. Many of those withdrawn or no-further-action accusations would also have been designated as false if they had been investigated.

This sounds like you are suggesting (correct me if I'm wrong) that many or most withdrawn and 'no-further-action' accusations are actually false, which is not a fair conclusion to draw from the information you presented. It also seems to claim that cases designated as withdrawn or 'no further action' are not investigated, which is also not accurate. Specifically, the paragraph you shared mentions many of the reasons victims will withdraw or not seek further action: 

• Withdraw: 
  • "when complaints were withdrawn, suspects were more likely to be current or former partners of the complainant" Victims of domestic violence are often in immediate physical danger when attempting to leave or report, or are extremely financially limited due to their situation in an abusive relationship. Following through on an accusation could pose more danger to the victim than withdrawing.
  • Investigating sexual assault cases can be extremely invasive, and often re-traumatize the victim.  Cases can then take years to prosecute and involve cross-examining the victim. Some victims withdraw to avoid this process.
• No-Further-Action: This is a police designation indicating they decided not to investigate for any number of reasons, including lack of forensic evidence, uncooperative witnesses, lack of resources, or a belief that they will not be  able to successfully prosecute. None of these directly imply a false claim. 
  • At least in the US, the reports often initially come through Patrol officers who typically don't receive the specialized training in responding to sexual assault victims (e.g., trauma-informed training) that investigators working in domestic violence or sex crimes units do, and may be less likely to take reports seriously. 
  • "Cases that resulted in "no further police action" were more likely to involve younger victims" Sexual assault is rarely the only issue in cases with young victims, and gathering evidence is made more complicated by this.
  • "victims who had consumed alcohol or other drugs near the time of the offense" In addition to the potential for police prejudice against these victims, there's also the consideration of whether or not a jury is likely to find a victim's story compelling. This indicates a judgement on the potential for a successful prosecution, not on whether a crime was actually committed. 

I agree that this is a "different story," just not one indicating a secret cache of false claims hidden by statistics. Instead, it's a depiction of the extremely messy process of reporting, investigating, and prosecuting these cases.

I can't access the full article from which you pulled that abstract, but I used a number of the resources below in my response. I hope you find them informative:

https://www.ojp.gov/pdffiles1/nij/grants/252689.pdf (Particularly the qualitative interviews where police describe lack of resources, lack of training for patrol officers)

https://www.uml.edu/News/stories/2019/Sexual_Assault_Research.aspx (short summary of the above)

https://www.london.gov.uk/sites/default/files/london_rape_review_final_report_31.7.19.pdf (seemingly a similar type of report to the abstract you included, but with context around the findings)

https://www.psychologytoday.com/us/blog/finding-new-home/201810/rape-allegations (on some reasons people don't report, which seem relevant here)